Reduction of data from spectral analysis



United States Patent [72] Inventors Charles D. Prater Pitman; James Wei, Princeton, NJ. [2]] Appl. No. 854,698 [22] Filed Sept. 2, 1969 [45] Patented Dec. 29, 1970 [73] Assignee Mobil Oil Corporation a corporation of New York Continuation of application Ser. No. 155,236, Nov. 20, 1961, now abandoned which is a continuation-in-part of Ser. No. 49,921, Aug. 16, 1960, now abandoned.

[54] REDUCTION OF DATA FROM SPECTRAL ANALYSIS 16 Claims, 18 Drawing Figs.

[52] U.S.Cl ..235/151.35, 235/180, 235/150.1 [51] Int. Cl .1. G06g 7/34, GOln 21/00 Primary Examiner-Eugene G. Botz AttorneysAndrew L. Gaboriault, Oswald G. Hayes and James F. Powers ABSTRACT: In spectrographic analysis, the combination of m of the n available peaks of a spectrogram is automatically selected for use in concentration determination with reduced susceptibility to error. Functions are generated representing the determinants of sets of equations relating the concen'trations of the constituents and the peaks in the spectrogram. One of the larger, or the largest determinant, identifies that set of equations to be used in determining error concentration. The search for the determinants of greatest magnitude includes generating simplexes derived from all combinations of the simultaneous equations to identify the m X m determinants of greatest magnitude.

PEAK 2 COMPONENT 2 PATENTEUUEBZSIQTB SHEET 3 BF 6 PEAK I Z SPACE FIG. I2

COMPONENT I FIG. 8

HEIGHT OF PEAK 2 Io'o HEIGHT OF PEAK I HEIGHT OF PEAK 3 I00 HEIGHT OF PEAK I X- COMPOSITION PLANE 2N QB 2 52 DO 50 a CONCENTRATION OF COMPONENT I IOO- Y-PEAK HEIGHT N PLANE Q I LL 8 2 A O I 2 Lu 3:

HEIGHT 0F PEAK I I.o- 2 FIG. IO 0 l- I 2 2 Lu 0: 2 H D 2 IL L) Q g Q Q a CONCENTRATION OF COMPONENT I PATENTEU UEEZS 19m SHEET '4 [1F 6 -FIG.7

m zqwm mo :65:

FIG. 6

HEIGHT OF PEAK I CONCENTRATION OF COMPONENT I REDUCTION OF DATA FROM SPECTRAL ANALYSIS CROSS'REFERENCES TO RELATED APPLICATIONS SUMMARY OF THE INVENTION This invention relates to spectral analyses such as carried out by use of mass spectrographs and the like and more particularly to the identification in spectrographic data of a particular set of data from the available sets thereof which will yield an analysis of least error.

In a more specific aspect, the invention relates to the selection from a set of simultaneous equations of a particular subset, in number equaling the number of constituents of a mixture to be tested where such subset is characterized by having a determinant which is a maximum.

In a further aspect, the invention relates to the identification and utilization of a particular subset of simultaneous equations involving data selected from different analytical techniques.

In the study of processes presently used in industries such as petroleum industry and in the development of new processes, detailed and accurate analysis of complex mixtures of hydrocarbons and other chemicals are needed. For example, the success of studies of kinetic mechanisms of cracking and hydrocracking reactions depends strongly upon reliable analytical results. Analytical procedures must be rapid and inexpensive to provide the large quantity of data necessary to such studies.

Mass spectrometry and gas chromatography are methods that approach these requirements. However, mixtures too complex to be analyzed by the ordinary techniques may be encountered.

Often failure in such analysis may be attributed to human limitations in the evaluation of the error transformation characteristics of the sets of simultaneous equations ordinarily employed in the analysis of data from mass spectrometry. Such limitations restrict the usefulness of the method. On the other hand, gas chromatography often suffers from a lack of resolution between certain components of a complex mixture.

Even with an understanding of the error transformation characteristics of sets of simultaneous equations, it is difficult, if not impossible, to remove the human limitations in their evaluation. Even the greatest reliability that can be achieved often is inadequate when a single analytical technique is used. In order to overcome the foregoing limitation, applicant has devised methods for identifying the most desirable set or sets of data to minimize error transformation characteristics of the simultaneous equations in which such data are employed. As a result, appreciably more reliable analyses are obtained for very complex mixtures than heretofore possible.

The present invention is most useful in connection with spectrographic analyses and particularly mass spectrographic analysis of mixtures, the constituents of which are known but concentrations of which are to be determined.

Such a problem is present in monitoring the output from a chemical processing unit in which process variables and/or feed stocks may change and data as to the resultant change in the make-up of the product is necessary. For mixtures, the constituents of which are known, spectra may be obtained for pure samples of each constituent so that the contribution of each constituent for unit concentration in a mixture may be determined. With such information, data from the mass spectrogram in which peaks Y Y,...Y,-...Y,, occur may then be employed for the solution of simultaneous equations of the following form:

where Y, is the peak height for the mass number a is the height of the peak for a unit concentration of the pure component i X i is the concentration of the component of the mixture,

However, in general, there are more peaks in the spectrum than constituents in the mixture, and thus in the use of all such peaks the problem is overdetermined. It is desirable to reduce to a minimum the computations involved and at the same time to utilize the data from the spectrogram which is more reliable. A brute force approach to the solution of equations such as equation set 1) would be to solve the set of equations for all combinations of the n-peaks in the spectrum taken in subsets of m. However, even in moderately complex mixtures the possible subsets become astronomical in number. For example, in a spectrogram obtained from the analysis of a lO-component mixture in which there are 20 peaks from which to choose, there are 184,756 possible subsets of 10 equations available. systemically to solve each of the above subsets is wholly impractical.

Experienced personnel working with such data often develop the art whereby a fairly good subset of peaks can be selected by the application of relatively simple tests. However, the chances of choosing one of the better subsets, even by the experienced personnel, are small. The present invention therefore is directed to a systematic method for the selection of the best or one of the best subsets of peaks in a spectrum in which peaks exceed in number the constituents of the related mixture.

One subset of data may be considered better than another subset merely by reason of the fact that some peaks are more susceptible than others to error. Errors which may be involved in peak heights generally fall into one of two classes. The first are errors which are proportional to the peak height. The second class of errors are those which are independent of peak height and include background noises, contaminants remaining in sample holders from previous runs, uncertainty as to the base line on the resultant spectrogram, etc. For the purpose of the present description, it will be assumed that the magnitude of possible error in a given peak height is known and may be considered to be input data the same as the contributing factors a 11, etc. of equation set l In the present invention attention is directed toward minimizing any amplification of such error in transforming data involving peak heights Y Y ...Y, to data indicative of concentration of the constituents of the mixture through solution of equations such as equation set (1). More particularly it has been found that error amplification in such transformation is minimum in that subset of equations having a determinant greater than the determinant for any other set of equations involved spectral data from the same mixture.

In spectrographic analysis where a mixture, the components of which are known but concentrations thereof are related to maximums in the resultant spectrums and are to be determined from the solution of simultaneous linear equations of the form of equation set (1), there is provided a method for identifying the subset or subsets of data which will minimize error. The method comprises generating physical representations of the determinants of said simultaneous equations with the n maximums in the spectrum taken in sets of m equal to the number of said constituents.

Physical representations of the concentration of each component in the mixture are then generated from relationships in the subset of said equations yielding the largest determinant.

In a more specific aspect, the method involves identification of the better set of spectral peaks by normalizing with a preselected normalizing function each of the elements of the determinant in each equation of the equation set l Physical representations of the determinant of the normalized matrix corresponding with the n peaks taken in subsets of m, where m is equal to the number of the constituents in the mixture, are then generated.

In one aspect, normalization of the elements of the determi-' nant involves obtaining a representation of the ratio of each element divided by a function representative of corresponding peak height. In a further aspect, normalization involves modification of the magnitude of each element in a-given row in the determinant so that the sum of the elements in each rowequals unity. In a further aspect, normalizationinvolves modi-- f'ying each element of the determinant bya function-proportional to the probable error in the height of thecorrespondihg spectral peak. In accordance with a further aspect of the invention, a solution of a selected subset of equations is provided where at least one of said subsets involves data obtained from an analytical procedure different from the data obtained for the remaining equations;

For a more complete understanding of the'present invention V and for further objects and advantages thereof, reference may now be had to the following description taken in conjunction with the accompanying drawings in which:

FIG. 1 is a block diagram illustrating spectral analysis;

FIG. 2 illustrates a system for identifying a-preferred.determinant;

FIG. 3 illustrates the further portion of the:system of FIG. 2'

which serves to compute, from the data represented by a preferred determinant, concentrations of components of a mixture;

' FIGS. 4 and 5 illustrate related peak height and concentration data for a two-component mixture;

FIGS. 6 and 7 illustrate peak height and concentration data for a two-component, three-peak mixture;

FIGS. 8-10 illustrate the effect of probable error in the FIG. 18 illustrates a system for identifying a third set of data functions for use with the pair identified by the system of FIG. 17;

Referring now to FIG. 1, there is illustrated in block form a mass spectrograph 10 in which asample to be analyzed is placed. The output circuit of the spectrograph 10 is connected to a recorder 11 from which there issues a spectrogram, a record chart 12, on which there appears. a curve 13 having peaks therein plotted along a scale representative of mass number M. The curve 13 may be considered to be a spectrum representative of the dissociation patterns of the constituents of the sample in spectrograph 10. Mass spectrographic analysis is based upon the assumption that, for mixtures, the peak height at any given mass number is determined by the additive:

contributions from individual components of the mixture as given by the linear algebraic equation set (1) above; In the spectrum of FIG. 1 peaks 1-4 prominentlyappear above the background. Equation set (1) may be written more con veniently in terms of matrix algebra as follows:

Y=AX' where Y stands for the matrix (vector) Y1. X stands for the matrix (vector) and A? stands for the matrix (1 1.0 g a (11 G21 G 2 (l (1 a (Z3 a a 0 1 a a a where the values of a a etc., are representative of the contribution from individual components to a given peak. It will be'noted that equation (2) is of general form.

Many of the-terms of equation (2) are' not required in analysis'of the four-peak spectrum of FIG. 1'. p

More particularly in theanalysis of the spectrum 13 of FIG. 13,,itwilliberassumedithat'tlie spectrum is known to result from a sample'in unit l0'which is a two-component mixture. In such case, only the first four of the equations of equation set (3) would be considered, and the problem would be to select from thesixpossible pairs of-the first four equations the particular" pair which willyieldthe most accurate results. The equationswhich could be-ruse"d?to determine the concentration of'th'e': two components represented by the spectrum 13 may be wrii-- ten as follows:

1 u i'i' 12 2 2 21 1 22 2 3 31 1 I 132 2 4 41 1 M2 2 In accordancewith'the present invention the pair of-equations selected from equation set (3) is the pair having a determinant of greatest magnitude. More particularly assume that the A-matrix of equation set (3) for the two-component, fourpeak system as'illustrated by spectrum 13 is as set forthin the following table:

TABLE. I*.--NORMALIZED A-MAI RIX FOR A TOW-COMPONENT, FOUR- PEAK SYSTEM Components Peak No.:

The sum of the values of each row of the A-matrix in table I is equal to unity, the A-matrix having been so normalized. Such normalization is based upon the assumption that each of the two components present in the sample in unit.l0 is of unit concentration. Stated" otherwise, the assumption is that the sample'is an equimolar mixture. Other modes of normalization may be employed. and maybe desired, and accommodation therefor'is made as hereinafter described.

The inventionaapplied to the present example involvessearching the normalized A-matrix of table I for the determinant of greatestvalue and than having identified theset of:

equations havingthe greatest determinant generating a'physical representation of the concentration of each componentin the mixture dependent upon the peakheights corresponding to such determinant.

The forgoing may more readily be understood by referring, to the drawings-and particularly FIGS. 2 and 3 wherein a system is illustrated forsearching the A-matrix for the greatest determinant and for providing a solution for the simultaneous equations having such determinant.

More particularly, the signal represented by spectrum 13' may be reproducediandianalyzed for peak heights by unit 15- where, as a result: voltages appear on four output terminals. representative of the peak heights Y,, Y Y and Y The output terminals-on unitl5 will then be connected (connections' not being shown')'to.terminals similarly identified in-FIG'. 3'. Suitablevoltage'rsources 16 are provided for each ofi'the ele- DETERMIN ANT SEARCH UNIT In FIG. 2 a voltage N, representative of a preselected normalizing function is applied to an inverter 20, the output of which, the reciprocal of N,, is applied to the first input terminal of a multiplier 21. A voltage a from source 16 is applied to the second input of multiplier 21 so that a voltage channel 22. The reciprocal of the voltage N is also applied to one input terminal of multiplier 23. A voltage a is applied to the second input terminal of multiplier 23 so that a quotient representing the quotient appears on the output voltage appears on channel 24. Similarly through multipliers 25, 26, 27, 28, 29 and 30 quotients 2a a 1 1 1a a & 2, 2, a, 3, 4, 4 32, 33, 34, and 36, respectively.

Channels 22, 24, 31-36 are connected to selected terminals of a ganged switch 37. Each section of the ganged switch 37 has an armature associated therewith which is actuated under the control of a driving motor 38, a mechanical linkage 39 extending to each armature. The armature 40 of section 37a is connected to a first input terminal of a multiplier 42. Armature 43 of switch section 37b is connected to the appear on channels 31,

second input terminal of multiplier 42. Similarly, armature 44 nected by way of resistor 50 to the input terminal 51 of a cathode-ray oscilloscope 52. The output of multiplier 46 is connected by way of an inverter 53 and resistor 54 to the input terminal 51 of oscilloscope 52.

The terminals of the sections 37a--37d of switch 37 are connected to channels 22, 27 and 31-36 such that armatures d0, 43, 44 and 47 in switch position No. 1 evaluate the deterrepresentative of that detenninant. When the motor M moves the switch armatures to the No. 2 switch terminals, the deteru n] n s: toltage being applied to terminal 51. Similarly, each of the four remaining determinants is evaluated and voltages success ively are applied to the oscilloscope 52 representative of said determinants.

minant and apply a voltage to terminal 51 minant is evaluated, a representative As illustrated on the oscilloscope screen, steplikevoltages appear, clearly indicating that the determinant shaft 62. Shaft 62 is supported parallel to the face of the oscilloscope 52 and is coupled by way of linkage 63, reducing gear box 64 and linkage 65 to motor 38. Motor 38 has one terminal connected to battery 66. Battery 66 is connected by way of a normally closed switch 67 to the second terminal of motor 38, completing the power circuit. Switch 67 is operatively associated with a latch 68 and an actuating solenoid 69. The

photocell 60 is connected by way of an amplifier 70 to solenoid 69. In response to a signal from photocell 60 an energizing voltage is applied to solenoid 69 to open switch 67 and at the same time lock the switch arm in open position as by means of latch 68. Manual reset of the switch 67 is then required before motor 38 can again be energized.

In order to identify the determinant of maximum value photocell 60 is mounted behind a baffle 71 also carried by follower 61. Baffle 71 has a fine slit 72 across the face thereof. The sweep of oscilloscope 52 is contracted as to be entirely within the limits of the sensitive area of the photocell 60 and is positioned to coincide with the track of the photocell as it moves down across the face of the oscilloscope. Follower 61 is initially positioned with the slit 72 at the upper edge of the oscilloscope face. Switch 67 is then closed so that the switch 37 is cyclically rotated to produce, in succession, voltages at terminal 51 each representative of the determinants subsets of equation set (3) dependent upon the matrix of table I. At the same time the coupling from motor 38 to the threaded element 62 slowly moves the photocell 60 downward until the upper tip of the excursion of the cathode-ray beam for the maximum determinant is in registration with slit 72. When such registration occurs (i.e., with the function on the scope representative of the determinant of subset No. 6) solenoid 69 is energized to open switch 67, latching it open. The armatures of switch 37 all, in the example given, then rest in contact with "at s:

By this means all combinations of the mamxffi twocomponent, four-peak system have been searched and the best set of data for analysis purposes has been identified.

DETERMINATION or CONCENTRATION The sections of switch 37, shown in FIG. 2, are coupled by linkage 39 to additional sections 37e-37p to select input functions suitable for computing the concentrations of x, and

the No. 6 terminal which yields the determinant x as related by the third and fourth equations of equation set The determinant solution for the above equations may be expressed as follows:

ai az aa u To carry out such computation, the terminals on each of the sections 37e--37p are connected to the matrix voltage sources 16 as the legends indicate (actual interconnections having been omitted). The armatures of sections 372 and 37f are connected to the input of a multiplier 100, the output thereof being connected by way of resistor 101 to the input of multiplier 102. The armatures of sections 37g and 37h are connected to the two inputs of multiplier 103 whose output in turn is connected to unit 104 which reverses the sign of the voltage output from multiplier 103. Unit 104 is connected by way of resistor 105 at the first input of multiplier 102. Resistors 101 and 105 are to be taken as representative of an algebraic adding network. The signal from multiplier 1 voltage representative of the numerator is applied to the first input terminal of multiplier 102. The determinant of the equations of equation set (4), the denominator of equation 5 is produced by applying voltages from the armatures of sections 37! and 37] to multiplier I06 and from the armatures of sections 37k and 371 to a multiplier 107. The voltage from multiplier 107 after change of sign in unit 108 is added to the volt age from unit 106 through resistors 109 and 110. The sum is then applied to inverter 111, whose output is connected by way of conductor 112 to the second input of multiplier I02. Inversion of the determinant voltage in unit 111 and multiplication with the voltage representative of the numerator as 'applied to unit 102 results in an output voltage applied to meter 113 which is representative of the concentration x of the first constituent in the mixture.

equations formed by selecting m'of the n available peaks. This is equivalent to writing the equations in the form;

. t lmytn xm= illy! bmafym '15 calculated from the given values of the a s by matrix inver- In a similar manner a voltage is developed from multipliers 120 and 121 and applied to multiplier 122 which voltage is representative of the numerator of equation 6. The detenninant voltage is applied by way of conductor 112 to the second .input terminal of multiplier 122 after inversion in unit 111 so that the product voltage from multiplier 122 is a voltage representative of the concentration x of the second constituent. The latter voltage is applied to meter 123.

Linkage 39 coupling the armatures of all of the sections on switch 37 serves to apply to multipliers 100, 103, 106, 107, 120 and 122 the proper voltages for the solution of equations 5 and 6. Thus, given the values of the peak heights and the proper values of the A-matrix, the sources such as source 16 may be preset for application to designated terminals on the sections of switch 37. Switch 67 may then be closed. When switch 67 is opened upon identifying the determinant of greatest magnitude, meters 113 and 123 will thenprovide an indicationof the concentration of the two constituents in the mixture.

The system illustrated in FIGS. 2 and 3 has been employed for illustrating application of the present invention to a relatively uncomplicated'problem. Subsets of equations having elements in each subset equal in number to the number of constituents in a mixture were searched to identify the determinant of maximum value. Following "this, the data corresponding with such determinant were then employed for determining the concentration of the components of mixture under analysis. The analogue system illustrated in FIGS. 2 and 3 is wholly automatic in its operation and carries out method of the present invention.

It will be readily recognized that more sophisticated systems may, with greater efficiency, handle the above problem while carrying out the self-same function embodied in the operation of the system of FIGS. 2 and 3. To illustrate, in the two-components, four-peak system switch 37 having four terminals per .section is adequate to search all possible combinations of six things taken two at a time. However, substantial extension beyond this system may be unduly complicated, rendering it unwieldy In contrast, thetlexibility of a digital computer lends itself to the solution of problems of the type above described and in most instances represents an instrumentality preferred for the carrying out of the method of the present invention.

The detailed search for the better determinant and the related better set of equations though involving considerable computation and equipment whether done by analogue or digital means is justifiable where repeated analysis of mixtures of the same constituents but in different proportions is to be performed. Having once identified the peaks in the'spectrum to be employed, thereafter the only variations in the system will be those necessary to accommodate data representative of the variations in the height of the selected peaks for the successive samples.

As above indicated, the values of the a s are determined by measuring spectra of the pure compounds. From the observed peak heights, Y of the mixture and values of the a s from the pure compound spectra, the values of x s can be determined provided there are at least as many peaks as there are components in the mixture. The linear algebraic equations obtained must be independent. As above described, the value of 7 the xfs are determined by solving the set of m linear algebraic sion procedures or, in terms of matrix algebra.

Where B stands for the matrix bi} i; bi}

, bi... bi... b...

The inversion of the A-matrix as indicated by equation set (7) and equation (8) in itself for large matrices is an involved: process as compared with the solution of a single set of sim ul tion of the values of the constituents as indicated in equationset (7). Thus, one would employ a normalized A-matrix to identify the subsets .of data to be employed and then invert the coefficient matrix of the selected subset to obtain the B-matrix of equation (8 It should be emphasized that the problem is not necessarily the selection of individual peaks in the spectrum having the least error but rather the selection of a set of peaks which, when errors are present, yields a composition least sensitive to those errors. For example,'with a good choice of peaks, a 1 percent error in peak height could yield a 1 percent error in composition, while a poor choice of peaks in which a l percent error in peak height is present could yield percent or greater error in composition.

The foregoing has dealt withthe relatively simple case of a two-component, four-peak mixture. In order to extend this method to multicomponent systems, the following analysis will be helpful.

GEOMETRICAL INTERPRETATION OF THE x "ALGEBRAIC SOLUTION 1 Letting x, be the concentration of component 1 and x .be

the concentration of the component 2 in some convenient units such as partial pressure in a vaporized sample,a mixture 1 thereof may graphically be represented by a point, P, in the composition plane shown in FIG. 4. Points on a straight line from the .origin and passing through P represent mixtures of the same relative icomposition differing only in the total quantity of the mixture. The intersection of the line 130 with v the line AB gives the composition in terms of mole fractions, i.e., points on the line AB correspond to a total pressure of l micron. Thus, for this specific example the point P represents;

a mixture containing 0.4 microns of component 1 and '.0.4 microns of component 2. or 50 mole flertent each For this composition the height, Y,. of peak I, and the height, Y of peak 2 are given by These peak heights can be represented graphically by the point P on the peak height (Y) plane shown in FIG. 5. Point P in the x plane, FIG. 4 corresponds to the point P in the Y plane, FIG. 5, through the action of the mass spectrometer 10, FIG. 1. Points along the line 'A correspond to difierent amounts of pure component 1 and points along the line 0'B' correspond to different amounts of pure component 2.

It is important to note that the peak height space differs from the composition space in that a restricted portion of the positive quadrant is used to represent all possible mixtures of the two components. That is, any mixture of components I and 2 must yield values of peak heights which fall in the region of the positive quadrant bounded by the lines OA' and O'B. In practice the point P is measured experimentally and then the position of the corresponding point P is determined from the data relating peak height to concentration.

PROPAGATION OF ERRORS IN THE ALGEBRAIC SOLUTION There will now be discussed the effects of experimental peak height errors on the calculated composition. Suppose that the absolute value of the combined peak height error 0', at P has a maximum of 25 percent of the peak height and that the combined errors obey the equation, a}: u fi+ d' g Under these conditions the sets of possible experimental peak heights are found within the circle of radius 0-,, about I" as shown in FIG. 5. Each point within this circle corresponds to a point in the composition space and in particular the points having the locus transform into a tilted ellipse in the composition space.

The ellipse shown in FIG. 4 was calculated from equation (4) using b values calculated from the a s. Several sets of corresponding points are given in table II and are shown in FIG.4and FIG. 5.

TABLE XL-CORRESPONDING POINTS IN PEAK HEIGHT AND COMPOSITION SPACES Mole fraction composition Coordinates Point Coordinates x1 x:

The second reason is that the error circle in the peak height space is distorted into an ellipse in the composition space. Points 3 and 4 clearly illustrate that peak height errors equal in magnitude but different in direction may lead to two widely different values of the calculated composition, one that corresponds exactly to the true composition, the other in error by a large amount.

When peaks 1 and 3 are chosen rather than peaks 1 and 2,

the peak height and composition spaceare as shown in FIGS.

6 and 7. The o',,=25 percent error circle about P is again shown and it is readily apparent that the error ellipse in the cpmposition space is much smaller than in the previous examp e.

The value of the reciprocal of the determinant is now only 1.25. Clearly as above noted in the solution afforded in FIG. 2, peaks 3 and 4 are even better since their determinant is unity and consequently no error magnification occurs. Also for this case there is no distortion of the error circle.

The normalized determinant, then, is a measure of the overall error characteristics of a given set of peaks and the better matrices are the ones with the largest determinants.

Stated differently, since the area of the triangle O'A'B' is equal to one-half the value of the determinant, the smaller the restricted area of the peak height space, the greater the effect of a given error in peak height on the calculated composition. This conclusion can be generalized to space with more than two dimensions where the areas become m dimensional volumes where error sensitivity of a matrix is formulated in terms of this m dimensional volume formed by the vectors and appropriate lines (planes) connecting their ends. The larger this volume the greater the reliability of the matrix.

NORMALIZATION In the example shown in FIGS. 4-7 an unrealistic error tolerance was used for the sake of clarity; also, the example involved a special case in which the components were present in equal amounts and the peaks were assumed to have the same relative errors, i.e., 25 percent. For this case the shape of the error area in the y space was a circle. In general, however, the error area is elliptical.

A further feature of the determinant method for selecting data least sensitive to error magnification is a comparison of the error area to the restricted area in the peak height space. The smaller the ratio of these areas the better the peak selection. To illustrate why the normalization procedure is justified, consider a case involving two components and three peaks. The A matrix for this system is given in table III.

Peak No.:

It is assumed for this example that all peaks have the same percentage error.

FIG. 8 gives the y space diagram for peaks 1 and 2 and FIG. 9 gives that for peaks 1 and 3. Note that area OAB of FIG. 8 is exactly one-third as large as O'AB' of FIG. 9 since the coefficients for peak 2 are one-third those of peak 3. For any given mixture of the two components, however, peak 2 is one-third as large as peak 3 and therefore the ratio of the error area to the total restricted area is the same in both cases. When the yspaces are mapped back to the composition plane (x-space) it is found that the error areas are identical. This is shown in FIG. 10. Thus it is clear that there is no reason to select one peak in preference to another on the basis of peak height alone. The value of the determinant of the A matrix, however, measures the actual area of the triangles and would favor the larger peak height. In order clearly to indicate the better data, each row of the A matrix is divided by a value typical of the corresponding peak height y in order to normalize it. The peak height employed to normalize a given row may be the actual peak height measured or it may comprise the peak height typically present in spectra for a plurality of samples of a given .mixture. In the former case, normalization procedure would imply a different set of row normalization values for each different mixture of the two components. In the case illustrated I in table .I a normalized A-matrix was used in which the'rows were adjusted so that the sum of the elements in a row was equal to unity. As previously'noted, this is equivalent to the assumption that each component is present at unit concentration, i.e., that normalization is on the basis of an equimolar mixture.

Note in FIG. 2 the normalization factors N,N., were preset to those values such that the rows of table I would be normalized to unity. Where values typical of the peak heights are to be employed, the terminals N N N and N may be connected to terminals Y',','Y Y and Y, of "FIG. 1 so that norl .malization would be dependent upon peak height.

An additional feature may be incorporated into the normalization procedure when the peaks considered have different relative errors. The modification necessary to accommodate such variation may be understood in terms of the preceding example of two components andthree peaks, table III. Assume that the error in peaks one and two are 25 percent of the measured peak height but that the error in peak three is only 12.5 percent. Now a comparison of the ratios of error ellipse area to total restricted area in the y space shows that the ratio for the system involving peaks one and two is twice as large as that for the system involving peak one and three. Thus peak three would clearly be indicated as the proper peak to use with peak one. To take into account this possibility one must normalize by dividing each row by the percent error expected for that peak, i.e. (for equimolar mixtures) peaks having one percent error would be normalized to one and those having two percent error would be normalized to one-half, etc.

This operation in FIG. 2 would involve modifying the value of normalizing factors N,N, by presetting them in dependence upon the relative errors involved in peak height determination.

i In the foregoing there has been presented the method of identifying sets of data which are susceptible in the least degree to error magnification. A normalized A-matrix having the maximum determinant is by the present invention identified through a systematic search of the possible matrices. Even with the substantial savings in time and the resultant increase in reliability of the analysis through identification of the largest determinant, the amount of work required to evaluate all possible determinants in the search for the best matrix may be prohibitive even in moderately complex analyses. While the search methods above outlined and illustrated by several examples constitute a substantial improvement, faster search methods can be employed while embodying the present invention, such methods resulting in the identification of one of the larger determinants, if not the largest, so that for even the most complex analyses the present invention finds immediate application.

SEARCH FOR MATRIX HAVING THE LARGEST DETERMINANT A more rapid search method has its origin in the geometrical interpretations above given in connection with FIGS. 4- -10. The introduction to such faster search method will now to presented in connection with FIGS. 11-15. The identification. of a specific determinant by either the foregoing techniques or those now to be explained is the result of a syste- 'matic search through possible combinations and can be relied upon as being one of the better determinants where in complex systems there may be several or many that may fall in a given range, any one of-which might be more advantageously 7 employed that any of thedeterminants falling outside that range. Thus,-it may not be necessary to identify the single determinant of greatest value but only to identify one of the determinants falling within the category of better determinants. In order to explain the systematic search method adopted for complex systems, it will be more convenient to use a-terminology different from that employed in FIGS. 4- -'l0, where a criteria of maximum area in y-space was adopted. In contrast, it will be desirable to employ a different set of parameters, i.e. a z-space. The relation between this Z space and the y space can be seen in FIGS. 11 and 12 using a two-component, two-peak system. In the-.yspace, the axes correspond to the peaks l and 2 .and the vectors, a, correspond to components 1 and 2. Vector q was,previously denotedby *O'A' and vector o by O'B..I n, the z space theaxes correspond to the components and the vectors to the peaks. In ithe 2 space the vectors will be designatedas l3 vectors. In the y space the coordinates of the a vectorsare given by the columns of-the A matrix while in the z-space the coordinates 5 idfthe-Bwectorsaregiven'by the rows of the A matrix. The triangles formed by the a and B vectors in the y and 1 space respectively have the same area since the value of the determinant is unchanged by interchanging the rows and columns. For the search procedure the z space has the advantage that all n peaks in a single m dimensional space may be examined.

The nature ofithe search procedure can be illustrated by reference to the three-component, sixpeak system specified in table IV. I

TABLE rvJ-A MATRIX FOR 3 COMPONENT, 6 PEAK SYSTEM Assuming an equimolar mixture that all six peaks have one percent error, the normalized coefficients have the values presented in table V.

TABLE V.-NORMALIZED A MATRIX FOR 3 COMPONENT, 6 PEAK SYSTEM Component Peak No.2

In this system three mutually perpendicular axes are employed to represent the components and B vectors in the first octant to represent peaks. The six ,8 vectors in 2 space are shown in FIG. 13. In three dimensions, the problem is to find the largest tetrahedron formed by three of these vectors since this is equivalent to finding the matrix with the largest determinant. This is accomplished by building up the volume, one vector at a time, starting with a given vector. Each pair of B vectors defines a plane in the three dimensional space. When the ends of each pair of vectors are connected by a straight line, triangles are formed. The triangle formed by B, and B is shown as a shaded area in FIG. 13.

' The evaluation of the area of analogous triangles in a two component system is carried out by evaluating the determinant of the normalized values of the a s. In the three component system the rows corresponding to peaks 1 and 6 are:

but since a determinant must have the same number of rows as columns, the area of the triangle cannot be easily calculated. However, the coordinate system 2 2'2, and z in FIG. 13 can be rotated so that two of the axes lie in the planedefine 'd by ,8,

are the coordinates of OB. Actually the rotation is carried out so that A coincides with one of the axes and thus 7, 0 and the area of the triangle is simply 'YuX 722'- 2 This procedure is carried out for each pair of vectors. The areas of triangles formed by vector 1 and each other vector are given in table VI. The largest area is that formed by vectors I and 6. i i

TABLE VI.-AREAS OF SOME TRIANGLES IN THE 3 COMPONENT, 6 PEAK SYSTEM Areas of Vectors: triangle 1, 2 3033 The triangle formed by vectors 1 and 6 will now be used as a starting point for forming a tetrahedron by combining them with one of the remaining vectors. Combining vectors 1 and 6 with one of the remaining vectors, x gives the array:

u I112 t:

n s: w 0x1 ax; 0x3 10 A determinant can be formed from (10) whose value is proportional to the volume of the tetrahedron formed by vectors 1, 6 and x. The values obtained for the determinants formed by vector l, 6 and all remaining vectors are given in table VII.

TABLE VII.-SOME DETERMINANT VALUES IN THE 3 COMPONENT, 6 PEAK SYSTEM Values of determinants The combination l, 5, 6 has the largest value and represents the largest tetrahedron starting with vector 1.

It is possible that a larger tetrahedron may be formed using a set of three vectors that does not include 3,. Consequently, this procedure is repeated using vectors 2, 3, 4, 5, and 6 in turn as the starting vector. The largest value produced by each of the six calculations are given in table VIII.

TABLE VIIL-VoLUMES 0F TETRAHEDRA rommn USING VARIOUS stars or MASS VECTORS Starting Vector:

nants, instead of 184,756. However, when applied to systems with a large number of components, the largest volume may dimensions. The area or volume thus enclosed will hereinafterbe referred to generically as a simplex. In order to find the desired simplex, one of the peak vectors from a spectrogram, properly normalized, is selected as a starting point. The next step is to determine which of the remaining vectors taken with the first forms the largest simplex. Thereafter a third vector is selected from those remaining which, when taken with the first two, forms the largest tetrahedron. Next, using this tetrahedron, a fourth vector is identified which forms the four dimensional simplex with the largest volume and so on. This procedure is repeated using each of the n vectors as the arbitrary first vector. Considerable saving in effort is thus achieved.

The simplexes formed in building up the final m dimensional simplex are described by rectangular matrices. For the p dimensional simplex, p, m, the matrix will be:

n 12 lm 21 22 2m The rectangular matrix 11 can, by rotating the coordinate system until p of the coordinate axes lie in the subspace of the p simplex, be conveniently represented in the following form.

DI D nm an an (1 m pzwn 11 00 QnQ12---Q1m zt zz 0 Q2tQ22--- m L31 L33 L33 0 0 nl nz ps pt- Q'aQ'a. 0;...

Pl i=1 (13) To apply the step-by-step process, the rows of the m X n matrix system are considered to be the vectors. That is:

.H- INK-Isle where A,, A A A are the n vectors in the m dimensional space. One of the A vectors is selected as the starting point and designated as A. The coordinates of the end of this vector is I X m matrix. The Gram-Schmidt orthogonalization method is used to rotate the coordinates, so that:

Next for each of the remaining A vectors the values of L L and the Q2, '8 are calculated so that:

where x signifies any one of the remaining vectors. The vector with the largest value of L is selected from this series of calculations and designated 11' These two vectors are combined with each of the remaining vectors in turn so that:

where again x signifies each of the remaining vectors. The vector which gives the largest value of L 3 is selected.

This building up procedure is continued until an m X m matrix has been selected. This matrix has the largest determi- I nant that can be selected by this method beginning with a given vector. This procedure is repeated beginning with another vector and so on through the entire set of n vectors. The largest determinant obtained from this set belongs tothe matrix with the best error propagation that can be selected by this method.

The above method can be easily refined to give a faster computing technique. Let the vectors be labeled A A A,,. One of these vectors A will be chosen as the starting point.

:and

Next, all the remaining elements of AP to A,, will be resolved to components parallel to Q,, and discarded, and to components perpendicular o Q1, designated as A5.

The vector A with the largest value of L is chosen for the second vector and and aw) 1 i l This procedure is repeated till m vectors have been selected. The procedure requires considerably less computation effort than the first one.

These procedures will not always select the largest determinant possible. For instance, when the six vectors of a 3-D system are displayed, Fig. 16, in a regular hexagon, triangle 135 is larger than triangle 124. But with these procedures starting from point 1, point 4 will be chosen next and the largest triangle 135 will never be chosen. However, the matrix selected is certainly one of the better ones if not the best.

Referring now to FIGS. 17 and 18, there is illustrated a system for carrying out a progressive search from a relatively simple two dimensional simplex to N-dimensional simplexes. The system is illustrative of the family of systems suitable for determinant search or simplex search in data from a spectrogram of a three-component, six-peak mixture of the type above discussed in connection with table V. It will be helpful first, however, to set forth a mathematical description generally as was employed for'equati on (3)f but expanded to encompass the following three-component, six-peak data expressed in the same terms as equation l From equation (3a) the simplexes to be searched may be tabulated in the form of the following group:

( n 12 1a 21 22 2s '31 sz aa (3b) '41 '42 43 51 s: ss n 62 53 The system of FIG. 17 is adapted to utilize data represented by group (3b) to determine the two-dimensional, three component simplex having the largest value. Stated otherwise, the system of FIG. 17 will identify the two rows in group (3b) which bound two sides of the triangle having largest area. This involves solving successively the simplex formed by rows l' and 2; then a second simplex formed by rows 1 and 3; a third, by rows 1 and 4, a fourth, by rows 1 and 5 and a fifth, by rows I and 6. Thereafter the combinations of each of rows 3-6 taken with row 2 are evaluated. In this manner all combinations of the groups (3b) are evaluated and that simplex of two three-dimensional vectors giving the largest numerical value is identified.

The area of the triangle bounded on two sides by the two three-dimensional vectors corresponding with rows 1 and 2 of equation (3b) may be represented as follows:

The 14 remaining combinations may similarly be expressed. The system of FIG. 17 is adapted to evaluate each of the fifteen possible combinations of the rows 1 to 6 of the simplex group (3b). The system operates to search such combinations cyclically to produce an output indication, the maximumwf which is selected to indicate the pair of rows which give the ia u largest area. Thereafter, the pair of rows thus selected is employed in sampling all of the possible combinations of the rows of group (3b) to determine the three by three simplexof greatest magnitude.

Referring now to FIG. 17, voltage sources are adjusted ,or selected in magnitude to represent the values (1 -11 of group (3b) and are connected (in the manner illustrated) to the 15 terminals in a first switch section 215. The switch 215 is-provided with a switch'arm 215a which is driven by way of a mechanical linkage 213 from a motor 38 sequentially to complete a circuit between the first input 2300 of multiplier 230 and each of the voltage sources connected to terminals on the switch 215. A plurality of switch sections 2l6226 are provided. Voltages are applied to the terminals of switch sections 216-226 as indicated. Armature 216a is connected to the second inputs 2305 of multiplier 230. Similarly armatures 217a and 2180 are connected to the two inputs of a multiplier 231; armatures 219a and 220a are connected'to two inputs of a third multiplier. 233; armatures 221a and 222a are conv nected to multiplier 234; armatures 223a and 2240 are connected to multiplier 235; and finally. armatures 225a and 2260 are connected to the input terminals of multiplier 236.

The output of multiplier 230 is applied by way of an adding network 240 to a squaring unit 241 and thence by way of resistor 242 to an output terminal 243. The output of multiplier 231 is applied to an inverting unit including multiplier 244 having a unit negative signal applied to one of the input terminals so that the output therefrom is the inverse of the product of the voltages appearing at the switch armatures 217a and 2180. The output of multiplier 244 is then applied by way of the adding network 240 to the squaring device and thus there is applied to output terminal 243 a voltage representative of the first element of the simplex (3c).

In a similar manner, the voltages applied by way of switch armatures 219a and 220a, 221a and 222a are combined and applied by way of resistor 245 to output terminal 243 to develop a voltage representative of the second element of the simplex (31). Finally, the voltages from switch armatures 223a, 224a, and 225a, and 226a employed through multipliers 235 and 236 to develop a voltage which is applied by way of resistor 246 to the output terminal 243 representative of the third element of the simplex (3c).

The sum of the output voltages representative of the simplex (3c) appearing at the output terminal 243 is then applied to a measuring unit 250 so that the magnitude thereof may be indicated. In addition, the output terminal 243 is also connected to terminal 51 which corresponds with the input terminal S1 of the sensing unit 52 of FIG. 2. It will be noted that the motor 38 is the same as the motor 38 of FIG. 2 so that the automatic homing operations above described in connection with FIG. 2 may be incorporated in the more complex system of FIG. 17. More particularly, the motor 38 drives the switch armatures 215a-226a cyclically to all of their respective voltage terminals on switches 215-226 so that all of the possible combinations forming the different simplexes from the rows in group (3b) may be sampled. Thus a voltage will appear on the indicating unit 250 which varies cyclically as the motor 38 drives the switch armatures 2l5a-226a. The switches 215-226 have been illustrated in a linear form for simplicity only. Conveniently, they may comprise rotary switches of the type illustrated in FIG. 3 to facilitate the cyclic operation desired.

It will be understood that many of the elements of FIG. 2 including the searching device 52 and their associated elements have been omitted from FIG. 17 merely for purposes of simplicity. However, they perform the same function with reference to the portions of the system illustrated in FIG. 17 as is performed in the search system of FIG. 2. By so operating, the motor 38 will be deenergized when the switch armatures 215I-226a are in contact with a set of terminals on switches 215-226 yielding the highest voltage on indicator 250. This voltage is representative of the simplex of greatest magnitude. Stated otherwise, the two three-dimensional vectors thus identified bound two sides of a triangle of the greatest area of any combination of the vectors represented by the group (3b).

As previously indicated in the discussion of table V above, the simplex involving rows 1 and 6 of group (3b) describes the triangle of the largest area. Thus in the system of FIG. 18 data representative of rows 1 and 6 are employed to search the remaining combinations to determine the three by three" simplex of greatest magnitude. More particularly, the volume encompassed by three three-dimensional vectors may be determined from the solution of the following determinant:

Equation (3d) may be more explicitly stated as 70 follows:

The system of FIG. 18 automatically evaluates the volume such as expressed in equation (3e) for each of the six determinants possible from equations (3a) by taking rows 1 and 6 together with the remaining rows taken one at a time. The section 260 of FIG. 18 serves to evaluate the first element of Equation (3e). The second section 261 serves to evaluate the second element of equation (3e) and the third section 262 serves to evaluate the third element of equation 3e). More particularly, voltages representative of terms 1123 11 3 of equation (3e) are applied to a first switch 266 and are applied successively to a first input of multiplier 267. The voltage representative of the term a is applied by way of conductor 268 to the second input of multiplier 267. Voltages representing terrns a ra 52 are applied to terminals in a second switch bank 269 and thence to one input of a second multiplier 270. A voltage representative of term a. is applied to the second input of multiplier 270 by way of conductor 27]. The output voltage from multiplier 267 is inverted as in unit 272 and added to the output voltage from multiplier 270 in the network 273. The latter voltage is then multiplied in multiplier 274 with a voltage representative of term a The product is then squared in unit 275 and applied to an output terminal 276. The voltage thus applied to terminal 276 is proportional to the first element of equation (3e), i.e.,:

2 m ss 1 In a similar manner voltages representative of the second and third elements of equation (3e) are applied to the output terminal 276.

As in FIG. 17, switch armatures 266a, 269a, 280a, 281a, 282a and 283a are driven as by way of linkage 284 extending from motor 38. Also the output voltage appearing at point 276 is the sum of the voltages individually representing the three elements of equation (3e). The output voltage at point 276 is applied to an indicating unit 285 and also is applied to the input terminal 51 of a homing unit (not shown) of the type shown in FIG. 2. The elements of the homing unit shown more completely in FIG. 2 have been omitted from FIG. 18 solely for the sake of simplifying the description and the drawings. However, the function is the same such that switch armatures 266a, 269a, 280a-283a are operated in synchronism so that rows 2-5 of equation (3a) may be sampled with rows 1 and 6. In a search in data for a three-component, six-peak mixture the motor 38 will be stopped with the switch armatures in contact with the set of voltages applied to the system of FIG. 18 representative of the largest determinant.

Thus through the use of the systems of FIGS. 17 and 18 an N by M simplex is evaluated first for two vectors. After identifying the two vectors which encompass the largest area, the two vectors so identified are utilized with all the remaining vectors to identify the combination of three vectors bounding a solid of the largest volume.

Thus through the use of the systems of FIGS. 17 and 18 there is identified that set of data which is least likely to lead to error in determination of the concentration of the constituents 1a,, x and x; of equation (3a).

This generalized step-by-step search method may then be extended to more complex systems merely by adding the components necessary to evaluate the values of the various simplexes involved. For example, in a four-dimensional system the simplex formed by two four-dimensional vectors is as follows:

Equation (3g) is derived from the more general equation for the area defined by two N-dimensional vectors which may for the general case be expressed in the following form.

Having determined in a four vector system the area bounded by two four-dimensional vectors, the value of a simplex comprised of three four-dimensional vectors may be determined by solution of the following nature:

In the general case the value of three N-dimensional vectors may be expressed as follows:

li li lk 2 The expressions contained in the foregoing equations (30)- (3j) will indicate the general procedure for determining step-by-step first the area between the pair of N-dimensional vectors. After having determined such area, then the volume between three N-dimensional vectors is determined. Having determined the three yielding the largest volume, then the volume between those three taken with a fourth of the N- dimensional vectors is evaluated. Through such step-by-step search the set of vectors yielding the largest or one of the largest volumes in N-dimensional space is identified so that it may be certain that at least one of the best sets of the data is identified for computations of concentration.

THE USE OF OTHER PEAK DATA IN MASS SPECTRA ANALYSIS Even using the best data that can be achieved in mass spectrometry, the results obtained may not be completely satisfactory since the determinant of even the best matrix for a particular analysis may be too small and consequently the analysis will be too sensitive to peak height errors. This is due to the close similarity between the fragmentation patterns of the various components. However, other analytical techniques yield data that may be expressed as linear equations. In accordance with the present invention provision is made for the inclusion of such data along with the mass spectral data so that more accurate analyses are obtained. For each method combined with mass spectrometry, one additional parameter c is employed to take care of the differences in sample sizes and instrument scale factors.

The sample sizes might be measured directly to give this parameter. However, sample sizes are difficult to determine and hold constant. Also, the value of the scale factors of the instrument tends to drift with time. For these reasons this technique is not useable to advantage. Alternatively, a known substance might be added to the sample to be analyzed as an internal standard provided it is well resolved in both techniques. The parameter could be calculated from this additional information each time.

In accordance with the present invention, neither of the above expedients is necessary. The parameter c can be obtained directly by the use of one additional peak for each method combined with mass spectrometry. In other words, for an m component system in which one additional method is to be combined with a mass spectrographic method, at least m 1 peaks are employed and the parameter c becomes another unknown on the same footing as the amount of each component. Thus, the data for each analysis are used as its own internal standard.

More particularly, when peak data for a given analysis are collected from two or more different analytical procedures, improved results often may be obtained by combining the groups of data for a single search. Such different procedures may include mass spectroscopy, gas chromatography, and/or infrared analysis. Even two separate runs-or tests using different operating conditions for a mass spectrometer may be considered to fall within the general area of combined methods. In the latter case, the use of different ionization voltages for two different runs would provide two sets of data which may be combined as will now be explained.

In initiating the search for evaluating the determinant of maximum value where two methods are being combined, there is first selected one peak from one of the two spectra, which peak from experience is known to be reliable in that it is one with respect to which it may be said that from any search the selected peak would be included in the determinant finally indentified. For example, in mass spectrographic analysis of refinery runs of gasoline, there appears a prominent peak in the spectrum which is due solely to the presence of benzene in the gasoline. This peak is readily recognized by those skilled in the art and is known as peak 78. Upon such selection or identification, one peak is eliminated from the search procedure, set to one side so to speak, while a systematic search as above described involving n linear equations relating the remaining sets of peak data would be carried out to identify the largest determinant of dimension m X m. Following this search, the set of peaks yielding the largest m X m determinant is then combined with the one peak previously set aside (as in the case of the peak 78 for benzene) to determine the value of the constant or parameter 0. With 0 evaluated, values of the concentrations of the various constituents in the mixture may then be determined.

The particular peak (for benzene) is not the only such peak that can be so employed but is given merely by way of example. In a mixture in which there is no benzene or the identity of the various peaks cannot be then a single peak in one of the spectra will be selected. A peak of desirable character would be one in which all of the constituents in the mixture contribute differently thereto. A poor choice of peaks would be one in which all constituents in the mixture contribute equally.

After the search as above discussed has been completed, the parameter c is evaluated from computations which include peak data for the one peak previously selected and set aside. The following mathematical formulation for evaluating 0 will illustrate a suitable set of operations. The following formulations are generalized to accommodate n peaks where there are p peaks of mass spectrometry data and n-p peaks of gas chromatography.

The procedure for evaluating c is based upon the least squares criterion expressed as follows:

l] m 2 E a; x,cY;) J=minimum with respect to both x x, and c.

The parameter c is then found by solution of the following expression:

i fi i ii i Yr il 3 1 (27) c has been computed there may equations:

After the value of be performed a solution for the x s in the as expressed in the square format.

In equations (28) and (29) peaks Y to Y,, are mass spectrometry peaks, and peaks Y, to Y,, are gas chromatography peaks. Where three methods are combined, the search would still involve m peaks with two peaks set aside for evaluating two c-parameters.

In equation (27) G is the matrix l-A(A'A)-A, where A is the coefficient matrix of equations (27) and 28) and A' is a transposed matrix formed by interchanging rows and columns of matrix A.

This method is designed to permit carrying out all of the computations that do not involve data from a particular sample. Certain matrices are evaluated for each given mixture and then relatively simple computations can be carried out for each particular sample of the mixture. Thus the determinant search need be made but once for any given mixture. The computations necessary for each particular sample are the evaluation of the G matrix equation (26) and the solution of equations (27) (29).

In this instance all peak data for a given analysis is collected including mass spectrographic analyses. gas chromatographic analyses, infrared analyses, etc. To the peak data thus assembled as in the form of input voltages or other physical representations thereof to a system such as generically represented in FIG. 1, a systematic search procedure is applied to different possible matrices preferably in a step-by-step process as above described for the registration of an indication of the determinant of maximum value. Such a search is made through determinants of order m l where m is the number of constituents of a mixture and where data from two analytical procedures are combined, or m 2 where three analytical procedures are combined. The search procedure is then carried out without regard to the parameter through the stepby-step process extending to the m units of the matrix. Thereafter one additional vector is added either from an evaluation of c as above indicated in connection with equation 26) or a vector dependent upon 0 representative of a median or representative value of the constant c.

As illustrated in FIG. I, the gas chromatogram 202 includes a curve 203 having peaks 5 and 6 thereon, the latter being recorded through the action of recorder 201 which is responsive to the output of the gas analyzing unit 200. Unit may then be employed to translate the spectrum 203 to peak height values Y,,+,, Y -ietc. so that there will be input functions for the determinant search unit of whatever form employed. Thus there has been illustrated and described a method and system for providing an indication of that determinant corresponding with a set of simultaneous equations which may be employed with confidence that the error in the solution of such equations will be minimal if not minimum. It will be appreciated that the system illustrated in FIG. 1 is an analogue system which is capable of carrying out the present invention. It should be understood, however, that analogue systems though capable of similarly operating for more complex mixtures do not represent the most desirable systems. Rather digital systems having grater flexibility in general will be found preferable.

While the invention has been described in connection with certain specific embodiments thereof, it will now be understood that further modifications will suggest themselves to those skilled in the art and it is intended to cover such modifications as fall within the scope of the appended claims.

We claim: 1. In spectrographic analysis where, from a given mixture of m constituents, spectral functions having peaks therein are obtained and wherein the relationships between the concentrations of said constituents and the peaks in said function correspond to relationships in a set of linear simultaneous equations, the system for selecting from said functions the combination of m of the n peaks therein least susceptible to error in concentration determination, where n is an integer greater than m, which comprises:

means for generating successive scalar functions each representative of a determinant for sets of equations corresponding with different sets of m of said peaks, and

means for determining that one of said scalar functions of greatest magnitude thereby to identify said combination of m of the n peaks least susceptible to error. 2. In spectrographic analysis where, from a given mixture of m constituents, spectral functions having peaks therein are obtained and wherein the relationships between the concentrations of said constituents and the peaks in said function correspond to relationships in a set of linear simultaneous equations, the system for selecting from said functions the combination of m of the n peaks therein least susceptible to error in concentration determination, where n is an integer greater than m, which comprises:

means for generating successive scalar functions each representative of a determinant for sets of equations corresponding with different sets of m of said peaks, and

means for determining that one of said scalar functions of greatest magnitude thereby to identify said combination of m of the n peaks least susceptible to error,

2. In spectrographic analysis where, from a given mixture of m constituents, spectral functions having peaks therein are obtained and wherein the relationships between the concentrations of said constituents and the peaks in said function correspond to relationships in a set of linear simultaneous equations, the system for selecting from said functions the combination of m of the n peaks therein least susceptible to error in concentration determination, where n is an integer greater than m, which comprises:

means for generating successive scalar functions each representative of a determinant for sets of equations corresponding with different sets of m of said peaks,

means for determining that one of said scalar functions of greatest magnitude thereby to identify said combination of m of the n peaks least susceptible to error, and

means for indicating the concentrations of said constituents from said combination of m of the n peaks least susceptible to error.

3. In mass spectrographic analysis where, form a given sample of material there is generated a spectrum function having peaks therein spaced along a mass scale with respect to which the relationship between concentration, contribution factor of each of the m constituents of the mixture and the magnitude of each of the n peaks in said spectrum is represented by a set of m linear algebraic equations and where n is an integer greater than m, the system of selecting for analysis a set of m peaks least susceptible to error in concentration determination which comprises:

means for dividing each said contributing factor for each peak by a normalizing function, means for successively generating from said contributing factors after their normalization a determinant function for each said set of peaks whose magnitudes are representative of the magnitudes of the determinant of each said set of equations, means for successively generating from said determinant functions output indications of the magnitudes of said determinants, and means for selecting from said output indications the determinant of greatest magnitude for identification of its equations and therefore the peaks least susceptible to error.

4. ln spectrographic analysis where a mixture, the components of which are known but the concentrations thereof are related to peaks in the resultant analytical spectrum and may be determined from solution of simultaneous linear equa- Yn= n1w1+ sm+ ni i+ nm m where Y, is the peak height for the i mass number, a is the height of the i peak for a unit concentration of the pure component j, :0, is the concentration of the j component of the mixture, where m is the number of components, n is the number of peaks, and n is always greater than in,

and where said determinant of each set of simultaneous equations comprises a matrix composed of m-vectors, and where a simplex is defined as an area encompassed by two of said vectors and as a volume encompassed by more than two of said vectors, the system of identifying the m peaks from which said concentrations may be determined with minimum error which comprises:

means for generating physical representations of a first family of simplexes derived from all combinations of said simultaneous equations taken two at a time, and

means for generating physical representations of families of simplexes of successively higher orders derived from combinations of said simultaneous equations taken in progressively increasing numbers to and including m wherein each member of each of said families of higher order includes the simplexes of larger values in the family of next lower order for identifying the m X m determinant of greater magnitude thereby to identify the simultaneous equations which include said peaks from which said concentrations may be determined with minimized error.

5. In spectrographic analysis where a mixture, the components of which are known but the concentrations thereof are related to peaks in the resultant analytical spectrum and may be determined from solution of simultaneous linear equations of the following form:

n ul 1+ n2 2+ ni a+ mwm where Y, is the peak height for the i mass number, a is the height of the i peak for a unit concentration of the pure component j, x, is the concentration of the j component of the mixture, where m is the number of components, n is the number of peaks, and n is always greater than in, and where said determinant of each set of simultaneous equations comprises a matrix composed of m-vectors, and where a simplex is defined as an area encompassed by two of said vectors and as a volume encompassed by more than two of said vectors, the system of identifying the m peaks from which said concentrations may be determined with minimum error which comprises:

means for generating physical representations of a first family of simplexes derived from all combinations of said simultaneous equations taken two at a time,

means for generating physical representations of families of simplexes of successively higher orders derived from combinations of said simultaneous equations taken in progressively increasing numbers to and including m wherein each member of each of said families of higher order includes the simplexes of larger values in the family of next lower order for identifying the m X m determinant of greater magnitude thereby to identify the simultaneous equations which include said peaks from which said concentrations may be determined with minimized error, and

means for producing from solutions of said identified simultaneous equations which includes said peaks giving rise to minimized error indications of the magnitudes of said concentrations in said mixture. 6. The method of determining with minimum error from the spectra of spectral analysis the concentration of the components of a mixture where the components are known and the concentration-determining peaks of the spectral analysis are present in number exceeding the number of said components,'which comprises in an automatic processing system the steps of generating physical representations in said system of the magnitudes of the coefficients of simultaneous linear equations defining the concentrations of said components as functions of the heights of said peaks of said spectral analysis, generating in said system from said physical representations of the magnitudes of said coefficients the magnitude of the determinant of a plurality of sets of said simultaneous equations, the number of equations of each of said sets being equal in number to the number of said components,

in said system comparing said physical representations of the magnitudes of said determinants of said sets of equations for identification of the set of said equations whose determinant has the largest magnitude, and

generating physical representations of the concentration of each said component of said mixture from said physical representations of the magnitudes of said coefficients of said set of simultaneous equations having said determinant of largest'magnitude and from said heights of said peaks included in said last-named set of equations.

7. In spectrographic analysis where a mixture, the components of which are known but the concentrations thereof are related to the height of the peaks in the resultant analytical spectrums and may be determined from solution of simultaneous linear equations of the following form:

Y =a x +a z a x a z where Y, is the peak height for the i" mass number, a is the height of the i peak for a unit concentration of the pure component j, x, is the concentration of the j component of the mixture, where m is the number of components, n is the number of peaks, and n is always greater than m,

the method of determining concentrations while minimizing error which comprises'in an automatically operable system the steps of generating in said system physical representations of the magnitude of determinants of sets of said simultaneous equations, the number of equations in each of said sets being equal in number'to the number of said components, and responsive to the condition in said system representative of the determinant of greatest magnitude generating physical representations of the concentration of each component in said mixture from relationships in said equations dependent upon the peaks in said spectra yielding said largest determinant.

8. In spectrographic analysis where, from a given mixture of m constituents. a spectral function having peaks therein in number greater than m is obtained and wherein the relationships between the Concentrations of said constituents and the peaks in said function correspond to relationships in a set of linear simultaneous equations, the method of selecting from said function the combination of the peaks in number equal to m least susceptible to error in concentration determination, which comprises in an automatic system the steps of applying to said system inputs whose magnitudes are representative of values of the coefficients of each said set of simultaneous equations, generating from said inputs successive scalar functions each representative of the value of the determinant for each of said sets of equations corresponding with different sets of m of said peaks, and

registering said scalar functions in relationship to the order of said sets of peaks to identify the scalar function of greatest magnitude.

9. ln spectrographic analysis where, from a given mixture of m constituents, spectral functions having peaks therein are obtained from at least p different analytical procedures, where p is greater than one, and wherein the relationships between the concentrations of said constituents and the peaks in said functions corresponds to relationships in a set of linear simultaneous equations, the method of selecting from said functions the combinations of m p 1 peaks therein least susceptible to error in concentration determination which comprises in an automatic system the steps of applying to said system input whose magnitudes are representative of values of the coefficients of each said set of simultaneous equations, generating from said inputs successive scalar functions each representative of the value of the determinant for each of said sets of equations corresponding with different sets of m pl of said peaks, and

registering said scalar functions in relationship to the order of said sets of peaks to identify the scalar function of greatest magnitude.

10. The method of claim 9 in which there is performed the additional step of selecting said inputs to said system to be representative of values of the coefficients of selected sets of simultaneous equations which include said peaks from one of said analytical procedures in combination with p- 1 additional peaks from the other of said analytical procedures for generation of said scalar functions.

II. The method of determining with minimum error from the spectra of spectral analysis of concentration of the components of a mixture where the components are known and the concentration-determining peaks of the spectral analysis are present in number exceeding the number of said components, which comprises in an automatic system the steps of applying as inputs to the system physical representations of the magnitudes of the coefficients of simultaneous linear equations defining the concentrations of said components as function of the heights of said peaks of said spectral analysis, said simultaneous equations being of the following form:

a is the height of the i peak for a unit concentration of the pure component j,

m is the number of components, n is the number of peaks,

generating from said physical representations of the magnitudes of said coefficients the magnitudes of the determinants of a plurality of sets of said simultaneous equations, the number of equations of each'ofsaidsets being equal in number to the number of said components,

in said system comparing said physical representations of the magnitudes of said determinants of said sets of equations for identification of the set in said equations whose determinant has the largest magnitude, and

generating in said system physical representations of the concentration of each said component of said mixture from said physical representations of the magnitudes of said coefficients of said set of simultaneous equations having said determinant of largest magnitude and from said heights of said peaks included in said last-named set of equations.

12. The method of claim 11 in which there is the added step of applying a normalizing function to said physical representations of the magnitudes of said coefficients.

13. The method of claim 11 in which there is the added step of modifying the values of said physical representations of the magnitudes of said coefficients by a normalizing function having a magnitude corresponding with the quotient of each coefficient in a given row divided by a function representative of the magnitude of the peak in said spectra related to said row.

14. The method of claim 11 in which there is the added step of applying a normalizing function to said physical representations of the magnitudes of said coefficients of said row thereof in each determinant to bring their sum approximately equal to unity.

15. In spectrographic analysis where a mixture, the components of which are known but the concentrations thereof are related to peaks in the resultant analytical spectrum and may be determined from solution of simultaneous linear equations of the following form:

. auZ -iaz ili and where said determinant of each set of simultaneous equations comprises a matrix composed of m-vectors where a simplex is defined as an area encompassed by two of said vectors and as a volume encompassed by more than two of said vectors, the method of identifying the m peaks from which said concentrations may be determined with minimum error which comprises in an automatic processing system the steps of:

applying inputs to said system whose magnitudes are representative of the values of the coefficients of each of said simultaneous equations, generating in said system from said inputs physical representations of families of simplexes of successively higher orders derived from combinations of said simultaneous equations taken in progressively increasing num- 

